Finite Group Databasebeta

← #224901

Subgroups — #224901

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36590C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic, complemented
3#21345C211yesmaximal, characteristic, complemented
4#9688C212yesmaximal
5#9688C212yesmaximal
6#9686C211yesmaximal, characteristic, complemented
7#9686C211yesmaximal, characteristic, complemented
8#8582C212yesmaximal, complemented
9#8579C211yesmaximal, characteristic
10#8225C212yesmaximal, complemented
11C4×#329C211yesmaximal, characteristic
12#2861C2212yes—
13#2859C2211yescharacteristic
14#2594—22no—
15#2582C411yescharacteristic
16#2538C411yescharacteristic
17#2525—22no—
18#2514C412yes—
19#2513C412yes—
20#2494—21no—
21#2421—21no—
22#2362—21no—
23#2355C411yescharacteristic
24#2354C411yescharacteristic
25#2347—21no—
26C4×#125C2212yes—
27C4×#124C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2212yes—
38#1767C2212yes—
39#1745C2212yes—
40#1745C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#330C2212yes—
44C2×#330C2212yes—
45C2×#330C2212yes—
46C2×#330C2212yes—
47C2×#329C2211yescharacteristic
48C2×#329C2211yescharacteristic
49C2×#329C2212yes—
50#1295C2211yescharacteristic
51#1295C2211yescharacteristic
52#1295C2212yes—
53C2×#125C2312yes—
54C2×#124C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97C2311yescharacteristic
69C2×#97C2311yescharacteristic
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×#97C2311yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76#401—22no—
77#400—22no—
78#398—22no—
79#397—22no—
80#397—22no—
81#396—22no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C411yescharacteristic
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#388—22no—
97#388—21no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—21no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#383—22no—
112#382—21no—
113#382C2×C412yes—
114#382C2311yescharacteristic
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C43C2312yes—
131C43C2311yescharacteristic
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#330—24no—
141#330—24no—
142#330—42no—
143#330—42no—
144#329—22no—
145#329—22no—
146#329—42no—
147C22×Q8C22⋊C411yescharacteristic
148C22×Q8C22×C411yescharacteristic
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161#124—21no—
162#124—21no—
163#124—21no—
164#124—21no—
165C22⋊Q8—44no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×D4—42no—
171#117—44no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220#97—21no—
221#97—21no—
222#97—22no—
223#97—22no—
224#97—22no—
225#97—22no—
226#97—21no—
227#97—21no—
228#97—21no—
229#97—21no—
230#97—42no—
231#97—42no—
232#97#4112yes—
233#97#4112yes—
234C4×C8—42no—
235C4×C8—42no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24—21no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4—81no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—81no—
333C22⋊C4—82no—
334C22⋊C4—81no—
335C22⋊C4—82no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—44no—
349C42—42no—
350C42—82no—
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C42H(5,𝔽2)11yescharacteristic
354C422−1+411yescharacteristic
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23C22×C22⋊C411yescharacteristic
363Q8—44no—
364Q8—81no—
365Q8—81no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—84no—
411C2×C4#41312yes—
412C2×C4#42112yes—
413C2×C4#42112yes—
414C2×C4C4×#4111yescharacteristic
415C2×C4#52011yescharacteristic
416C2×C4#54811yescharacteristic
417C2×C4#54811yescharacteristic
418C2×C4#55011yescharacteristic
419C2×C4#55011yescharacteristic
420C8—42no—
421C8—42no—
422C8—42no—
423C8—42no—
424C8—82no—
425C8—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22#234111yescharacteristic
434C22C2×#41311yescharacteristic
435C22#288911yescharacteristic
436C22C2×#42111yescharacteristic
437C22C2×#42111yescharacteristic
438C22#290111yescharacteristic
439C22#290111yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—41no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—81no—
455C4—82no—
456C4—82no—
457C4—81no—
458C4#153514yes—
459C2—161no—
460C2C2×#153511yescharacteristic
461C2#1768211yescharacteristic
462C2#1768311yescharacteristic
463C2#1768711yescharacteristic
464C2#2253711yescharacteristic
465C2#2255111yescharacteristic
466C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21345C2semidirect
3. #21345C2semidirect
6. #9686C2semidirect
7. #9686C2semidirect
8. #8582C2semidirect
10. #8225C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365902
#213452
#96882
#96862
#85822
#85792
#82252
C4×#3292